Here is a way, construct the equilateral triangle inside the shaded area connecting the centers of the circles and one point of intersection. You now know the area of the sector, given by (r^2/2)ß, ß = pi/3 and you can find the area of the triangle using pythagorean theorem and (base*height)/2

If you draw a line segment between the two points of intersection then two circular segments of equal area are formed.
Go to that webpage. Formula #17 gives the area.
But there is a problem: Their is your .
There is your .